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Malliavin Calculus and Its Applications Conference Board of the Mathematical Sciences CBMS Regional Conference Series in Mathematics Number 110 Malliavin Calculus and Its Applications David Nualart American Mathematical Society with support from the National Science Foundation Malliavin Calculus and Its Applications http://dx.doi.org/10.1090/cbms/110 Conference Board of the Mathematical Sciences CBMS Regional Conference Series in Mathematics Number 110 Malliavin Calculus and Its Applications David Nualart Published for the Conference Board of the Mathematical Sciences by the American Mathematical Society Providence, Rhode Island with support from the National Science Foundation NSF-CBMS Regional Conference in the Mathematical Sciences on Malliavin Calculus and Its Applications held at Kent State University, Ohio, August 7–12, 2008 Partially supported by the National Science Foundation. The author acknowledges support from the Conference Board of Mathematical Sciences and NSF Grant #0735352. 2000 Mathematics Subject Classification. Primary 60H07; Secondary 60H05, 60H10. For additional information and updates on this book, visit www.ams.org/bookpages/cbms-110 Library of Congress Cataloging-in-Publication Data Nualart, David, 1951– Malliavin calculus and its applications / David Nualart. p. cm. — (CBMS regional conference series in mathematics ; no. 110) Includes bibliographical references and index. ISBN 978-0-8218-4779-4 (alk. paper) 1. Malliavin calculus—Congresses. 2. Calculus of variations—Congresses. I. Title. QA274.2 .N828 2009 519.2–dc22 2009003082 Copying and reprinting. Individual readers of this publication, and nonprofit libraries acting for them, are permitted to make fair use of the material, such as to copy a chapter for use in teaching or research. Permission is granted to quote brief passages from this publication in reviews, provided the customary acknowledgmentofthesourceisgiven. Republication, systematic copying, or multiple reproduction of any material in this publication is permitted only under license from the American Mathematical Society. Requests for such permission should be addressed to the Acquisitions Department, American Mathematical Society, 201 Charles Street, Providence, Rhode Island 02904-2294, USA. Requests can also be made by e-mail to [email protected]. c 2009 by the American Mathematical Society. All rights reserved. The American Mathematical Society retains all rights except those granted to the United States Government. Printed in the United States of America. ∞ The paper used in this book is acid-free and falls within the guidelines established to ensure permanence and durability. Visit the AMS home page at http://www.ams.org/ 10987654321 141312111009 Contents Preface vii Chapter 1. The derivative operator 1 1.1. Hermite polynomials and chaos expansions 1 1.2. Derivative operator: Definition and properties 3 1.3. Derivative in the white noise case 9 Chapter 2. The divergence operator 13 2.1. Properties of the divergence 13 2.2. The divergence in the white noise case 15 2.3. Clark-Ocone formula 17 Chapter 3. The Ornstein-Uhlenbeck semigroup 19 3.1. Mehler’s formula 19 3.2. Hypercontractivity 20 3.3. The generator of the Ornstein-Uhlenbeck semigroup 23 Chapter 4. Sobolev spaces and equivalence of norms 27 4.1. Meyer inequalities 27 4.2. A continuous family of seminorms and Sobolev spaces 29 4.3. Continuity of the divergence 30 Chapter 5. Regularity of probability laws 33 5.1. Existence and formulas for probability densities 33 5.2. Regularity of the density 35 Chapter 6. Support properties. Density of the maximum 39 6.1. Properties of the support of the law 39 6.2. Regularity of the law of the maximum of continuous processes 40 Chapter 7. Application of Malliavin calculus to diffusion processes 45 7.1. Differentiability of the solution 46 7.2. Existence of densities under ellipticity conditions 48 7.3. Regularity of the density under H¨ormander’s conditions 49 Chapter 8. The divergence operator as a stochastic integral 55 8.1. Skorohod integral 55 8.2. Stochastic calculus for fractional Brownian motion 59 Chapter 9. Central limit theorems and Malliavin calculus 67 9.1. Central limit theorems via chaos expansions 67 9.2. Stein’s method and Malliavin calculus 72 v vi CONTENTS Chapter 10. Applications of Malliavin calculus in finance 75 10.1. Black-Scholes model 75 10.2. Computation of Greeks 76 10.3. Application of the Clark-Ocone formula in hedging 80 Bibliography 81 Index 85 Preface This monograph is compiled from the notes of a series of ten lectures given at the NSF-CBMS Conference on Malliavin Calculus and Its Applications at Kent State University, Ohio, August 7th to 12th, 2008. The Malliavin calculus or stochastic calculus of variations is an infinite-dimen- sional differential calculus on the Wiener space that has been developed from the probabilistic proof of H¨ormander’s hypoellipticity theorem by Paul Malliavin in 1976 (see the reference [25]). Contributions by Stroock, Bismut, Kusuoka, and Watanabe, among others, have expanded this theory in different directions. The main application of Malliavin calculus is to establish the regularity of the probability distribution of functionals of an underlying Gaussian process. In this way one can prove the existence and smoothness of the density for solutions to ordinary and partial stochastic differential equations. In addition to this main application, the Malliavin calculus has proved to be a powerful tool in a variety of problems in stochastic analysis. For example, the divergence operator can be interpreted as a generalized stochastic integral, and this has been the starting point of the development of the anticipating stochastic calculus. In the last years some new applications of Malliavin calculus in areas such as central limit theorems and mathematical finance have emerged. The purpose of these lectures is to introduce the basic results of Malliavin calculus and its applications. We have chosen the general setting of a Gaussian family of random variables associated with an arbitrary separable Hilbert space. Some of the applications are just briefly introduced, and we recommend the reader to look over additional references for more details. In the first three chapters we introduce the fundamental operators: the deriva- tive operator D;itsadjointδ, called the divergence operator; and the generator of the Ornstein-Uhlenbeck semigroup, denoted by L. Chapter 4 is devoted to proving the Meyer inequalities and the continuity of the divergence operator in the Sobolev spaces. The remaining chapters deal with a variety of applications of Malliavin calculus. First, in Chapter 5, we establish the general criteria for the existence and smoothness of densities for functionals of a Gaussian process. In Chapter 6 we discuss properties of the support of the law of a given Gaussian functional that can be proved using Malliavin calculus. Chapter 7 deals with the proof of H¨ormander’s hypoellipticity theorem. In Chapter 8 we discuss the use of the divergence as an anticipating stochastic integral with respect to the Brownian motion. This chapter also contains an introduction to the stochastic calculus with respect to the fractional Brownian motion, using techniques of Malliavin calculus. Chapter 9 presents some recent applications of Malliavin calculus to derive central limit theorems for mul- tiple stochastic integrals, and Chapter 10 describes some applications of Malliavin calculus in mathematical finance. vii viii PREFACE Finally, I would like to express my gratitude to Oana Mocioalca, Frederi Viens, and Kazim Khan for encouraging me to prepare these lectures and for organizing a very stimulating and interesting workshop. I would also like to thank all the participants for their helpful remarks and questions. David Nualart Bibliography [1] E. Al`os, O. Mazet and D. Nualart, Stochastic calculus with respect to Gaussian processes. Ann. Probab. 29 (2001), 766–801. [2] E. Al`os and D. Nualart, An extension of Itˆo’s formula for anticipating processes. J. Theoret. Probab. 11 (1998), 493–514. [3] D. Bakry, L’hypercontractivit´e et son utilisation en th´eorie des semigroupes. Lectures on probability theory (Saint-Flour, 1992), Lecture Notes in Math., 1581, Springer, Berlin, 1994, pp. 1-114. [4] D. Bell, The Malliavin Calculus, Dover Publications Inc., Mineola, New York, 2006. [5] F. Biagini, Y. Hu, B. Øksendal, T. Zhang, Stochastic calculus for fractional Brownian motion and applications. Probability and its Applications, Springer, London, 2008. [6] J. M. Bismut, Martingales, the Malliavin calculus and hypoellipticity under general Hrman- der’s conditions. Z. Wahrsch. Verw. Gebiete 56 (1981), 469–505. [7] J. M. Bismut, Large deviations and the Malliavin calculus, Progress in Math. 45, Birkh¨auser, 1984. [8] N. Bouleau and F. Hirsch, Dirichlet Forms and Analysis on Wiener Space,WalterdeGruyter & Co., Berlin, 1991. [9] Ph. Carmona, L. Coutin, and G. Montseny, Stochastic integration with respect to fractional Brownian motion. Ann. Inst. H. Poincar´e 39 (2003), 27–68. [10] P. Cheridito and D. Nualart, Stochastic integral of divergence type with respect to the frac- H<1 41 tional Brownian motion with Hurst parameter 2 . Ann. Inst. H. Poincar´e (2005), 1049–1081. [11] J. M. Corcuera, D. Nualart and J. H. C. Woerner, Power variation of some integral fractional processes. Bernoulli 12 (2006), 713–735. [12] L. Decreusefond and A. S. Ust¨¨ unel, Stochastic analysis of the fractional Brownian motion. Potential Anal. 10 (1998), 177–214. [13] E. Fourni´e, Eric, J. M. Lasry, J. Lebuchoux, P. L. Lions and Nizar Touzi, Applications of Malliavin calculus to Monte Carlo methods in finance. Finance Stoch. 3 (1999), 391–412. [14] B. Gaveau and P. Trauber, L’int´egrale stochastique comme op´erateur de divergence dans l’espace fonctionnel. J. Functional Anal. 46 (1982), 230–238. [15] Gross, Abstract Wiener spaces. 1967 Proc. Fifth Berkeley Sympos. Math. Statist. and Prob- ability (Berkeley, Calif., 1965/66), Vol. II: Contributions to Probability Theory, Part 1 pp. 31–42, Univ. California Press, Berkeley, Calif. [16] Y. Hu and D. Nualart, Renormalized self-intersection local time for fractional Brownian motion. Ann. Probab. 33 (2005), 948–983.
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